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Ca^UcLme WArtillerie de la Marine. 



Lieutenant United States Navy. 










The following is a translation of a memoir by P. Breger, Capitaine 
d'Artillerie de la Marine, published in the Mernoires Militaires et Sci- 
entifiqnes, 1877, to which is prefixed a deduction of the i)robability 
curve and the method of finding the probable rectangle. The deduc- 
tion of the probability curve is taken x^i^incipally from Merriman's 
Method of Least Squares. Without this previous knowledge the me- 
moir of M. P. Breger would be hardly appreciated or understood. 

If a gun be fired over a range a number of times under as nearly as 
may be the same circumstances of elevation, lateral train, charge of 
powder, and projectile, a certain difference will always be found in the 
results of the different rounds, both in range and in dcA^ation. As all 
these rounds were fired with the same degree of care, the results have 
equal weight, and as they are discordant, we can never be sure that we 
have found the absolutely true value of the quantity we seek 5 we must 
therefore be content with the most probable value, and determine within 
what limits it is probably correct. 


1. We give, therefore, by way of introduction, the definition and some 
of the first principles of probability. 

The word prohahility as used in mathematical reasoning means a num- 
ber less than unity, which is the ratio of the number of ways in which 
an event may happen or fail to the total number of possible ways. 
Thus if an event may happen in a ways and fail in h ways, and each of 
these ways is equally likely to occur, the probability of its happening 

is — — y' and the probability of its failing is -^ry* Thus probability is 

always expressed as an abstract fraction, and is a numerical measure 
of the degree of confidence which we have in the happening or failing 
of an event. If the fraction is 0, it denotes impossibility, if J it denotes 
that the chances are equal for and against its happening^ and if it is 1 
the event is certain to happen. 

2. Hence unity is the mathematical symbol for certainty. And since 

an event must either happen or not happen, the sum of the probabilities 

of happening and failing is unity. Thus if P be the probability that an 

event will happen, 1— P is the probability of its failing. 


3. If an event may happen in a ways and also in a' ways and fail in 
h ways, the probability of its happening is, by Article 1, — j — rri ) ^^^ 

since this is the sum of the probability of happening in a ways, and 
that of happening in a' ways, it follows that if an event may happen in 
different independent ways the probability of its happening is the sura 
of the separate probabilities. 

4. Let us now ask the probability of the concurrence of two iu de- 
pendent events. Let the first be able to happen in ai ways and fail in 
&i ways, and the second happen in «2 a-nd fail in ^2 ways. Then there 
are for the first event ai+&i possible cases, and for the second «2+&2; 
and each case out of the «i+Z>i cases may be associated with each case 
out of the <X2-f &2 cases, and hence there are for the two events («!+&]) 
(«2-|-&2) compound cases, each of which is equally likely to occur. In 
«i «2 of these cases both events happen, in hi h^ both fail, in ai h^ the first 
happens and the second fails, and in ^2 hi the first fails and the second 
happens. Thus we have for two independent events : 

ai a2 
Probability that both happen = (^^iji^^y-^^-p^^)- 

Probability that both fail = (^^^ft^/ (^^.j^)' 
Probability that the first happens and the second fails = 


(ai+hi) («^+&2)' 

Probability that the first fails and the second happens = 

^2 h 
{ai-\-hi) (a2+&2)* 

And the sum of these is unity, since one of the four events is certain to 
occur. Now, considering each event alone, the probability of the first 

happening is 77-07, and of the second .^^ , and since 

tti «2 Cll O2 

(ai+&i) (612+ &2) ai-\-h ^ «2+V 

we have established the important principle that the probability of the 
concurrence of several independent events is equal to the product of 
the separate x>robabilities. Thus, if there be four events, and Pi, P2, P3, 
and P4 be the respective probabilities of happening, the probability that 
all the events will happen is Pi P2 P3 P4, and the probability that all 
will fail is (1-Pi) (I-P2) (I-P3) (I-P4). The probability that the 
first will happen and the other three fail is Pi (1— P2) (1—1*3) (1— P4); 
and so on. 


5. Although it would seem at first sight that accidental errors could 
hardly be made the subject of mathematical reasoning, yet the very 
fact of their irregularity brings them under the laws of probability. 
Moreover we recognize that they must be subject to the following fun- 
damental laws of arrangement: 1st. Small errors are more frequent 
than large ones. 2d. Positive and negative errors (that is, quantities 
greater and less than the true value) are equally probable, and hence 
in a large number of trials are equally frequent. 3d. Very large errors 
do not occur, so that there is always a limit Z, siich that all the positive 
errors are included between and +?, and all the negative ones be- 
tween and— L 

These are the three fundamental axioms which, in connection with 
the principles of probability, are the foundation of the following rea- 

6. Therefore, different errors are not equally i^robable, for in a large 
number of observations a small error occurs more frequently than a 
large one, and hence has a greater probability, and an error greater 
than the limit I has a probability of or is impossible. Hence the 
probability of an error is a function of that error, so that calling x any 
error and y its i)robability, the law of probability of error is represented 
by the equation 

(1) .............2/=/(^), 

and will be determined if we can find the form off {x). 

7. If a number of rounds be fired from a gun under the same circum- 
stances, and the ranges be measured, and the mean range found, and 
the difference between each range and the mean be taken, these differ- 
ences will be called the errors. Suppose we take as the origin a point 
at a distance from the gun equal to the mean range and lay off the errors 
in range to the right or left of this point, according as they are positive 
or negative. The gun is then supposed to be to the left of the origin. 
If then corresponding to each error as an abscissa we draw an ordinate 
of a length proportional to the probability of that error, these ordinates 
and abscissas will be the co-ordinates of points of the probability curve, 

and its form will be somewhat like the figure from the axioms of Ar- 
ticle 5. 
We have called the difference between the ranges and the mean range 

the errors. The errors are really the differences between the ranges and 
the true range ; but this latter quantity we do not know. But, when 
the quantities are of equal weight, the arithmetical mean is supposed 
to be the most i)robable value, and therefore this is taken instead of the 
true value. 

8. Let the values of the ranges mentioned above be Ml, M2,M3 . . . M„. 
If z be the true value, the errors (z—Mx)^ (^— Mg) .... (^— M„) 
have been committed, and their respective probabilities are ^/i =/(2;— Mi), 
y.,=f[z—M^) .... 2/n=/('2^— M„); that is, if ii is the whole number 

of errors, and the error" (.2;— M2) occurs 1X2 times, y^z=z-l—f (z — M2); and 

we suppose enough ranges to be measured to exhibit the several errors 
in proportion to their respective probabilities. Now, the probability of 
committing all these errors is the product of their respective probabili- 
ties j or if P denote this quantity 

(2) . . . P=2/i2/2 2/3 . . . 2/.=/(^-Mi)/(^-M2) . . . /(^-M„). 

This probability P depends upon ^, the true value of the range, which 
is unknown. If we give to z various values we shall have correspond- 
ing values of P ; and in the imi)ossibility of finding the true value oiz 
we can only find its most probable value as given by the n ranges, and 
the most probable value of z is that for which Pisa maximum. To find 
the value of z which makes P a maximum we take the natural logarithm 
of each member of (2), giving 

(3) . . . logP=log/(^— MO+log/^0-M,)-f . . . +log/(^-MJj 

whence d¥_df(z-M,) df{z-^i;) ^ /(^-M, )_ 

Whence -^-y^^^^^+y^^^^^-\- - • • +;^^^:::m:)-^* 

and df{z-n^2) = 'P (^^M2)/(^-M2) d (^-Mg), 
and we have 


(4) . . -p- = ri)(^_Mi) + ^i^(^-M2) +^(^_MJ=0. 

Hence the value of z^ which makes (2) a maximum, is that which 
satisfies the equation 

(6) . . . ^(^-Mi) + ^i'(^-M2)+ #(^-MJ=0. • 

And this equation will furnish us with the most probable value of 0, 
provided that we can determine the form of the function #. 

Now, it is universall}' accepted as an axiom that with a number of 
quantities directly determined and of equal weight the arithmetical 
mean furnishes the most probable result; that is, the most probable 
value of z in the case under consideration is 

M1+M2+ M3+ . . . M, 

til ^— • 




This equation may be written 

?i^'=Mi+M2+M3+ . . . M,, 
(7.) .... (^_Mi) + (^— M2) + (^-M,)+ . . . +(^_MJ=0. 

That is to say, tlie arithmetical mean requires that the algebraic sum 
of the differences or errors shall be zero. 

Comparing now (6) and (7) we see that the symbol (P means merely 
the multiplication by a constant, since the value of z must be the same; 
(8.) . . . (?(^_Mi)4-^(^-M2) + , &c., =74^-Mi) + A:(^-M2) + , &c. 

Inserting in this the value of (P (2;— Mi), &c., from (5) we have 

(8a.) . . . _l f{'-^^) + ^/(^-M.) _ _ ^^,. 

= A;(^-Mi) + ^'(^-M2)4-&c. 

And since this is true for any number of ranges, it must be true for 
one, or two, or three ; hence the corresponding terms in the two members 
must be equal. If, then, x be any error, and y its corresponding prob- 
ability, so that 
(1.) y=f(x), we have from (8a) 

(9.) ....... -^/-{^^ =Jcx or ^^ =lcxdx. 

f(x)dx y 

Integrating this we obtain 

(10.) logyJ^'^+h'. 

Passing from logarithms, we have 
(11.) ?/=e*^^V =ce*^^'. 

Now y is to be positive, and is to decrease as x increases, either posi- 
tively or negatively ; hence the constant Ic is negative. Placing, then, 
^-7t=— /t^j we have 

(12.)' y=ce-^'^" 

as the equation of the curve represented in the figure. 

This equation satisfies the conditions imposed at the beginning of our 
investigation, for y is a> maximum when x=0, it is symmetrical with 
respect to the axis of Y, since equal positive and negative values of x 
give equal values for y; and when x becomes very large y is very small. 

9. To determine the constant c. 

Let Xi, X2, ^3 . . . if,, be a series of errors, Xi being the smallest, 
X2 the next following, and so on, the difference between the successive 
values being equal. Then, by Art. 3, the probability of committing one 
of these errors, that is, the probability of committing an error lying be- 
tween Xi and ^'„, is the sum of the separate probabilities ce-^'^^^^^ ceh'^xl^ 
&c., or if P' denote this sum. 

(13) . . . P 

'=c(^e-"'-;+e-'"1+ . . . +6-"'^:^ 

which may be written 
(13) P'=c2^e-^'^', 

which denotes the sum of the probabilities of all the errors from Xi to 
a?n, inclusive. P' denotes, then, the probability that an error lies between 
Xi and a?n. Now, if i denote the small interval between the successive 
values of x, and if our ranges are numerous enough so that x may be 
regarded as a continuous variable, i will be equal to dx-, then 

(14) ^ ~x e = i e ax, 

from which, by comparison with (13), we have 

(15) P'=i / e""% 

which expresses the probability that an error will lie between the limits 
Xi and Xj^. Now it is certain that the error will lie between— 20 and +00 , 
and as unity is the symbol of certainty, we have 

(16) 1 

from which the value of the constant c will be known as soon as we find 
the value of the integral between these limits. 

e-^-"^-" dx=2 j e-h'^' dx= -^ ; {a) 

.-. Ji I e-^'^' dx=zC or f e-^^ (i+^^) Mx = Ce-^' -j 

. f f e-^' (1+^') Mil dx=Q f e-^' dJi. (b) 
' 'J oJ Jo 

Let hx=z in (a) 


c-h^a+x-^) "1°° 1 

e-'.' (!+.') Mft=-^^;^^- = 2 (1+^) ' 


.-. by (b) and (c). 


Hence from (16) 

.-^ yjJL u — > — 

ih VtT 

(18) 1=^^ orc= 

Inserting this in (12), we have for the equation of the probability 
curve, or the law of the probability of error, 

(19) y=^U T-h e-^"^'. 

Inserting also the value of c in equation (15), we have 

(20) V'=^ r\-^'^'dx, 

which expresses the probability that an error will fall between the 
limits Xi and x^^. Also, since the integral between the limits —x and -^-x 
is twice the integral from —x to or from to -\-x^ we have 

as the probability that an error taken at random is between the limits 
—X and -\-x, or is numerically less than x. 

Now (19) is the equation of the probability curve, and the area be- 
tween the curve and the axis of x is 

Cydx=^ Ce-^'-'dx. 

10. Hence if (16) be multiplied by i it will be the total area of the 
curve, and if (21) be multiplied by i it will be the area between the lim- 
its —X and -\-x. Hence expressions (16), (20), and (21) are proportional 
to the areas of the probability curve corresponding to those limits, and 
if we regard the total area as unity, the partial area between the limits 
—X and -\-x will be a fraction given by P' in equation (21). Further, 
since errors are committed in j)roportion to their probabilities, these 
integrals and their corresponding areas are proportional to the number 
of errors which we should expect to find between those limits. If, then, 
we compute values of P' corresponding to successive numerical values of 
X in equation (20) they will be fractions proportional to the number of 
errors numerically less than Xj and at the same time express the prob- 
abilities of committing an error less than x. As, however, the constant 
h depends upon the precision of the ranges, and hence varies in differ- 
ent sets of rounds, we write equation (21) under the form 

(22) P^ = -4 I e-^'^ dhx= " / e-' dt 

and compute the values of P' corresponding to successive numerical 
values of hx or t by the usual methods of the integral calculus. These 
values are usually tabulated. 


11. Developing e-^^ into a series by Maclaurin's theorem, multiplying 
by dt and integrating, we get 

wliich is convenient for small values of t. 

If it=-j^, we find P'=0. 42839, and for ^= J, we find P'=0. 52050, 

whence by interpolation we have for P'=J, ^=0. 4769. 

Hence hx=0. 4769, or calling r this value of x, corresponding to 

P'=^, we write 

(24) r='-^, 

in which Ji is as yet unknown. 

12. Probable errors. 

By the probable error is usually meant the error the probability of 
committing which is J. Therefore r is the probable error, to know 
which we must determine 7i, the measure of precision of the result. 
Having taken the average ofn equally precise ranges MijMajMg, . , . M^, 
and calling Zq the arithmetical mean, we have 

^ _ M,+M,+M3 +M, . 

If z is the true value, and if Xi, x-zj Xs . , . . x^ are the errors, their 
values are 

(25) Xi=z—Mi, ir2=2;— M2, &c. 

Also, let Vi, t?2? ^3, &c., be the differences resulting from subtracting 
each range from the arithmetical mean, or 

(26) -yirrr^^o — Ml, t'2 = ^0 — M2, &C. 

If Zo were the true value of the quantity, or Zq=z, then the errors, x, 
would be the same as the differences, v. But as we can never be sure 
that Zo represents the true value, we can never determine the errors, 
Xi, X2, X3 . , . . x^. The values, Vi, V2, v^ . . . . v^, which are readily found 
in any particular case, we call residuals^ they should be carefully dis- 
tinguished from errors. 

13. The probability of the recurrence of any error, x^ is 

(19) . y=hi--^e-^'^% 

and the probability of the occurrence of the system of errors, x^, X2, 
X3 . . . . x^is the product of the separate probabilities. If, then, h is 
the same for each of the n observations, we have 

(27) . . . . P=2/i2/2 ijn = n'^i''7:-i^e-h'^^% 

where 2x^ denotes the sum 

^l^+^/ + ^3'+ +OOI 


Xow, iu this expressiou h is uDknown, and, further, we liave no means 
of finding" its exact value. But whatever be its value, P must be a 
maximum in order to give the most probable value of the range z. AVe 
are therefore led to conclude that for a giveii system of errors the most 
probable value of h is that which renders P a maximum. Differentiat- 
ing (27) with reference to Ji, and putting the first differential coefficient 
equal to zero, we have 


(29) ?i-2F:S'x2=0, 

from which 

(30) 2^=^^h}=^ 

We have thus the value of li in terms of ^j?^, which, however, we 
have no means of obtaining, since the errors 0*1, Xo^ X3 . . . . x^ depend 
upon the unknown true value z. If the number of ranges were infinite, 
2.1^ would equal ^r-, and, as the latter is determinate, h would 
be known. In a large number of ranges, therefore, the equation 

/«,-=—_,-- will always give a close approximation to the value of h. 

From (27) we see that Ix- must be a minimum when Pis a maximum 5 
calling, therefore, 2v- this minimum value of ^./'-, we will have 

(31) :^.i^ = :^v-' + ^'S 

in which Jc^ is a constant to be determined ; and then the value of 7t will 
be correctly given by insertiug in (30) the value of ^x- from (31). As, 
however, 2x^ cannot be exactly found, we cannot hope to find the exact 
value of A-, but must be content with determining an approximate one. 
Xow, theprobability of committing the systemof errors j?i, x-i^Xs .' . x^ 

(27) P = c'^ e-^'^^% 

or inserting for ^x'^ its value from (31) 

(32) . . . . V = c" €-^- (^ ^■'-{-^•■-) = c" e -^' ^^"- e-'^^^^. 

But c" e-'^- ^ ^"^ is by analogy the probability of committing the series 
of errors I'l ^2, &c., and is therefore an abstract number: calling it Ci, 
equation ^32) becomes 

(33) V = c^e-T^'^\ 

Hence, the law of the probability of any value J: is the same as that 
of an error x, AVe may regard, then, (33) as the equation of a curve of the 
same form as the probability curve, and we may show in the same 
manner that c^ = lu--^.. 


Therefore we may write (33) 

(34) :P = M7:-h€-h'^% 

which shows the probability P of a value 1c. 

Now, in this equation both h and Jc are unknown, and, as we have 
said before, we can only expect to determine their most probable values. 
The value of Ic^ although unknown is fixed and definite ; and hence we 
conclude as before that the most probable value of h will be that which 
makes P a maximum. Differentiating, then, equation (34) with reference 
to hj and placing the first differential coefficient equal to zero, we have 

(35) . . . ~^=i7c-h e-h'^'-2 hk^e-'''^' hi 7:-h=0y ' 

(36) . 1-2/^2^^ = 0, 

from which 

Therefore, from (31) 
(38) . ^^'=^^' + ^j^.^ 

as the nearest possible approximation. Since from (30) the value of 
2x^ is -^, we have 

^''^ w='''+w 

from which we find the most probable value of h in terms of the known 
sum of the squares of the residuals, or 

(^0) '-7w- 

Substituting this value in (24), we have as the probable error of a single 

(41) y=0. 4769. /_?^=0. 6745^/-^. 

14. The probability of the arithmetical mean is the probability of com- 
mitting the system of errors Vi, Vz, v^^ . . , v^, or 


and the probability that the true value of z is Zo-]-x/ is the probability 
of the system of errors Vi+x', V2-\-x'^ &c., or 

(42) . P/ = C" e-/i2 S(w+a;')2. 

Since i: {v-\-x'f=I v'^-\-2x' I v-\-n x'^, and 2' ^?=:0, this becomes 

(43) yj^^n ^-hH^V^+nx'^) , 


Hence we have 

(4-4) . . . Po : P^^ : : e - ^' - ^^ : e -^' i^v'^+nx'^) : : 1 : e - "^* ; 

that is, the probability of the error in the arithmetical mean is to that of 
the error x^ as 1 is to e - '* ^^ ^^ For a single range whose error is x', we 
have however ^0 '- yJ '• - ^ - 6-^'^". Comparing this with (44) we see 
that if h is the measure of precision of a single range, h ^/ n must be the 
measure of precision of the arithmetical mean Zq. Therefore Iiq denoting 
the measure of precision of ^o? we have 

(J:5) liQ=h Vn. 

Denoting by Tq the probable error of Zq, we have from (24) ho ro=hr= 


0. 4769: therefore 7'o=~-^, that is, the probable error of the arithmetical 

mean is equal to the probable error of a single range divided by the 
square root of the number of ranges. Hence 

m ro=4-=0.6745j_J!'' . 

Vn V n (n—1) 

15. Probable rectangle. 

If a number of rounds be fired from the same gun under exactly the 
same circumstances, and the ranges be measured, and the mean range 
found, then hj (46) the probable error of this mean may be determined; 
that is, the limits greater or less than the mean within which the i^rob- 
ability of a shot striking will be equal to the probability of its not 
striking. In like manner if the deviations to the right and left are 
measured, the lateral limits may be determined, v in this case repre- 
senting the difference between any deviation and the arithmetical mean. 
But a shot that strikes within the limits in the direction of the range 
may strike without the limits to the right or left. To determine, there- 
fore, the rectangle on the horizontal plane passing through the gun 
within which the probability of a shot striking is equal to the proba- 
bility of its not striking, we must proceed differently. 

Since the probability of two independent events occurring together 
is equal to the product of their separate probabilities, we may take any 
two numbers whose product is one-half and find the length of the side 
of the rectangle laterally corresponding to one of these numbers as a 
probability, and the length of the side in the direction of the range cor- 
responding to the other number as its probability, and the probability 
of a single shot striking within the rectangle would be one-half. Thus 
there may be any number of rectangles, described around the mean 
point of impact as a center, within which the probability of a shot strik- 
ing would be equal to the probability of its not striking. 

If we take the probabilities in the two directions as equal, and equal 
to the square root of one-half, the resulting rectangle is called the 


probable rectangle. Of course, the probable rectangle will vary in 
dimensions for different guns and different mean ranges, and its size 
will determine the accuracy of the gun for the mean range for which it 
is calculated. The smaller the rectangle, the greater is the accuracy of 

16. To determine the lengths of the sides of the probable rectangle, 
we let P'= VJ=--0.7071 in (22) 

p, 2 r^"" ,2 2,, 2 


and calculate the value of Jioo or t^ corresponding to this value of P' by 


For ^^=0.74 we find P'=0.70468 

and for M=0.76 we find P'=0.71754. 

Therefore for P'=0.7071, ]ix=0.7^3S 

(47) ... . . ^=-^-. 

From (24) 

, 0.4769 
r ' 

(^^) x=^'jr^^^r=lMr, 

and substituting the value of r from (41), 

(49) ;z?=1.56r0.6745 /^l, 

(50) ....... x=lM2^l^. 

V n — 1 

In (50) X is the error, positive or negative, measured from the mean 
point of impact, therefore 2x or 2a will be the length of the side of the 
rectangle in the direction of the range, when Iv^ means the sum of the 
squares of the residuals with reference to the range; and 2^1? or 2b will 
be the length of the lateral side of the rectangle, when Zv^ means the 
sum of the same quantities with reference to the lateral deviations. 
Calling these quantities Ivi and Uv2^, we have for the sides of the prob- 
able rectangle, 

(51) . 2a=2.wU^ 

(52) 36=2.104^^^. 


snr la 

Probabilite d'Atteindre Un But de forme quelconque. 

Par p. BEfiGEE, 

Capitai7ie WArtillerie de la Marine, 





The object of this study is the determination of the probability of hit- 
ting an object of any form. The employment of the ordinary methods 
fiirniwshes ns a solution of this i)roblem only in two particular cases : 
first, when we consider a rectangle of which the sides are parallel to the 
directions in which we measure the errors ; second, when we consider a 
circle having for its center the mean point of impact, which requires 
that the mean errors should be equal, a condition which is rarely real- 
ized; otherwise, one is stopped by the difficulties of the calculation. 
These methods, which are sufficient in the above cases, become inade- 
quate under the conditions of actual fire, for which the calculus of prob- 
abilities would be able to furnish the most valuable and positive informa- 
tion if we could always surmount these difficulties. Several times, and 
particularly in his Traite de balistique experimentale^ M. Helie has pro- 
posed, for practical use, replacing the probability curve, ordinarily rep- 
resented by the formula of Laplace, 

by a right line; these results are the development of this iug^enious 
idea; we thus avoid the almost impracticable management of a compli- 
cated exponent under the double integral sign. Without doubt this 
substitution causes an appreciable error, but it is extremely small, and 
has no practical value when we consider that from the nature of these 
calculations we cannot hope for an absolute precision; we can, moreover, 
by very simple means reduce almost to nothing this error, and this con- 
sideration will without doubt be of a nature to remove the scruples we 
might have against the employment of this empirical method, which, as 
we will show, affords a solution of a number of i)roblems unapi)roach- 
able by the usual methods. 

5782 2 17 




We cousider that the causes which produce the dispersion of the pro- 
jectiles around the mean point of impact are of two kinds; that the one 
acts all in the vertical direction, tending to raise or lower the projectile, 
while the other acts horizontally, tending to move it to the right or left: 
if, for instance, the target is vertical and has its center (Fig. 1) coin- 
cident with the mean point of impact, a particular point of impact P 
will be the result of firing a projectile subjected to the action of a ver- 


tical deviating cause, which produces the eri-or OA, and to a lateral de- 
viating cause, which produces the error OB. This is the preliminary 
hypothesis adopted; it is evident that we could make any number of 
others: admitting, for example, that the real directions of the errors are 
other than oy and ox^ or that there will be a greater number of them. 
It is true that these suppositions will be hardly justified; but we can 
imagine that there exists only a single cause of error acting indiffer- 
ently in all directions, or, better still, that the true law of dispersion 
holds at the same time of this last hypothesis and of the first; and each 
of these suppositions assigns a different value to the probability of hit- 
ting the point P ; if we adopt that of the vertical and lateral errors it is 
not because it appears certain, but, in the first place, the results to which 
it leads are sufficiently confirmed by experience; and, in the second 
l)lace, if we put to one side the two suppositions but little justifiable 
which we have mentioned above, it has the advantage of assigning to 
the probability of hitting the central part of the target its minimum 
value, and, since for this portion the suppositions the most oi^posite give 
results but little different, it errs on the safe side ; it is this which ex- 
])lains why it is to-day universally employed; but that only shows posi- 
tively that we can count upon it for an approximation. 




If we take a right line A B (Fig. 2), or rather two symmetrical right 
lines, A B and A B' for the curve of probability of errors measured 

along the axis x x from the point O, the probability of having an error 
oc OTx, to the right of the axis o s is equal to s dx ; in this case O B and 
O B^ measure the extreme errors, thus all the possible errors are com- 
prised between zero and OB, and zero and OB', the sum of all the s dx, 
or the area of the triangle ABB', should be equal to unity, the measure 
of certainty; hence the equation of the right line AB will be, calling m 

the extreme error o B, 


but it should be remarked that x can only vary between zero and m ; 
the probability of having exactly the error o c will be : 


that of not passing the error o c or of having an error comprised be- 
tween zero and +j? will be the integTal of this quantity between the 
limits zero and x, that is the area of the trapezoid O ADO, and, calling 
this P, we will have 

X X? 


The probability of not passing the error —x will be of the same value, 
and, if we consider the errors will be taken indifferently to the right or 
left of the point o, we will have for the probability of not exceeding the 
error x. 



2x of 

m iw 

We know that the mean error, which we will call y, is the abscissa of 
the center of gravity of the area O AB 5 we have then 






The area A is equal to J. We find easily 


and the same for the other side of the axis of s. Experience also shows 
that, in firing projectiles, the extreme error is a little more than three 
times the mean error j* but this is not of great importance ; that which 
is necessary, is that the probability of exceeding this error should be 
small enough to be neglected, and we cannot have any doubt in this 
respect. The mean error is ordinarily the quantity which we have 
given; replacing m by ^y in the value of P', we have the formula given 
by M. Helie : 

^--1^ 3r V 

but we will continue to use the quantity m, since it is more easily writ- 
ten, but will substitute y in any formula of interest. 

* The abscissa of the center of gravity of the area between the axis of x and one 
branch of the probability curve yz=-^ e , is fonnd thus : 

I e X dx 1 -1? x^ 

Jo ~'9hi -1 


-1? ^ ;j^ Vlt 

e ax . , 

^ 2h 


To eliminate Ti, we have, calling ?/o the value of y when x = 0; 

V Tt 


If, therefore, the abscissas of the centers of gravity of the triangle and the area 

inclosed by the curve are the same, we would have l=~-, and consequently the alti- 
tude of the triangle would be ^-, or a little greater than y^. Of course the value of 
Z is determined by the observations, and that of the altitude results from it. Since 
the mean error is equal to the absicssa of the center of gravity of the area between 
the curve and the axis of x, we have 





■y/ ji (mean error)' 
Since lir = .4769, when the probability is one-half, we have : 

r = .4769 -/^ (mean error) = 0.8453 (mean error) ; 
a value sometimes used for the probable error. C. A. S. 



The probability of hitting a particular point of impact is easily- ob- 
tained. The target can be in any position ; to fix the ideas we will sup- 
pose it horizontal. Through the mean point of impact o (Fig. 1) x)ass 
two rectangular axes oy and ox) upon the former measure the errors in 
range, and upon the latter the errors in direction 5 the point of impact 
P, of which the co-ordinates are x and y, results from the concurrence of 
two events : the error in direction x and the error in range y 5 the prob- 
ability pi of the error x is, as we have seen 

m—x ^ 

the probability jpa of the error y w^ill be 

n—y , 

in these formulas m and n represent the extreme errors in direction and 
in range ; the probability ]) of having at the same time the error x and 
the error 2/, or of hitting the point P, will be the ijroduct i)i|>2? and we 
will have 

(m—x) (n—y) , , - 

an equation in which x can vary from zero to m, and y from zero to n ; 
it is therefore apx)licable to the first right angle of the axes oy and ox, 
but it is easily seen that, in order to make it applicable to the other 
angles, it is only necessary to change the signs of ii? and y. 

In fact, to find the probability P of hitting any portion of the target 
inclosed by a curve, y=f (^), it is sufficient to integrate the expres- 
sion (3) between limits depending only upon y=f {x)^ and we will have, 
in the first right angle : 

(4) .... . ^=i^^2j dx£{m-x)xn-7j)chj. 

A similar formula derived by the ordinary methods would only gi^e 
a theoretical value, the integration indicated would be impossible 5 it is 
evident, on the contrary, that equation (4) is easily solved in a great 
number of cases ; the only difficulty would be when the integration with 
reference to the first variable introduced a function not iutegrable, 
which would hardly occur within the limits of practice. However, not- 
withstanding its simplicity, it requires some i)recautions which should 
not be lost sight of; the manner in which the subject will be presented 
in the following paragrai^h is of a nature to dispel all hesitation. 



AVhen we assign limits m and n to the two kinds of errors considered, 
the i)oint8 of imi)act cannot be outside of a determinate portion of the 
target; but when we use the formula of Laplace this is no longer the 
case, for the probability curve has then the axis of x for an asymptote^ 
In the actual case this is evidently an imperfection, for the errors can- 
not increase indefinitely ; with all others that we know, if the curve 
wiiich represents them admits a limit of errors, the portion of the target 
which contains all the points of imx^act will be a rectangle of which the 
center coincides with the mean point of impact, and of which the sides 
parallel to the directions of the errors will be double the extreme errors 
in length. 

Hence all points within this rectangle can be hit, for the co-ordinates 
of any point being less than the corresponding extreme errors, they can 
be considered as the result of one of the possible combinations of the 
two rectangular errors, varying from zero to their extreme value; with- 
out this rectangle, on the contrary, there can be no point of impact, for 
it would require one at least of the co-ordinates having a value greater 
than the extreme error in that directiou ; by analogy, in practice, we 
will call it the rectangle containing all the shots, or the rectangle of the 
extreme errors. 

Around the sides of this figure, and more particularly near the angles, 
the points of impact will be less numerous than towards the center, con- 
sequently there would be required a great number of shots to verify the 
fact that its form is really rectangular. In the hypothesis of the two 
deviations this deduction is exact; the size of the extreme errors causes 
some uncertainty ; but this is a point of no importance. 

Consider an infinite number of shots fired against a target which can 
be in any position, but which we will consider still horizontal, as in the 
last paragraph. Imagine each projectile reduced to the infinitely small 
dimensions dx^ dy, dz, of a rectangular parallelopiped standing upon the 
point of the target which it touches ; after firing, the i^rojectiles will be 
arranged in different numbers upon each element dx dy of the target; 
they may otherwise be considered as being suj^erimposed one upon the 
other, in such a manner that they will form a volume bounded by a cer- 
tain surface, of which the equation is z=:f (a?, y). 

The probability of hitting any point of the target will evidently be 
equal to the number of projectiles superimposed upon this point, 
divided by the whole number of projectiles ; but the volume v of the 
elementary vertical prism formed by the superimposed projectiles which 
have touched the given i^oint, and the whole volume Y will be in the 
same ratio as these numbers. The probability which we seek is then 

J v^, VV\ 


equal to ^. The height dz of the projectiles does not affect the result, 

for if it varies the volumes v and Y will vary it is true, but their rates, 
and consequently the probability, will not change; it will be of advan- 
tage to make Y=l; then the probability _p of hitting any point will be 
simply V ; or the volume of the elementary prism z dx dy. We can then 


We have already found : (3) 


from which we deduce 

(5) zj!!!^=^J^^. 

This is the equation of the required surface, or the surface of prob- 
aMUfy, a surface resembling, of course, the empirical form of the 
adopted curve of probability. 

If now we wish to find the probability of hitting any portion of the 
target, or, in other words, an object of any form, we must calculate the 
volume of a vertical cylinder having this portion for its base, and 
bounded above by the surface of probability, for this cylinder contains 
e^adently all the projectiles which, out of an infinite number of shots, 
would hit the object considered. Thus determining a probability is 
equivalent to calculating a volume; we have to go through the same 
operations, and they are the same as far as the calculation is concerned, 
but perhaps they are not the same as far as the facility of being misled 
is concerned. 



The surface of probability represented by equation (5) is a rectan- 
gular hyperbolic x)araboloid. This is readily shown by moving the axes 
parallel to themselves until the origin comes over the angle of the rect- 
angle of the extreme errors in the first right angle of the target ; the 
equation of the surface becomes then: 

(6) xy=m'^n^z. 

If z is constant, we have ii?2/= constant for the equation of the inter- 
section of all horizontal planes with the surface. This is the equation 
of a rectangular hyperbola. If either x or y is constant, we have the 
equation of a right line, as the equation of the intersection of the sur- 
face by a i3lane perpendicular to x and z, or y and z. 




Among all tlie sections which we can make of the surface of proba- 
bility, those made by planes parallel to the target have a certain inter- 
est ; they are called the carves of equal prohahiliti/. Thus, each of their 
points, being situated at the same distance from the target, has been 
formed by the superposition of the same number of projectiles, and 
consequently each of the horizontal projections of these points has the 
same probability. The equation of these curves is obtained by substi- 
tuting for z a constant value A in the equation of the surface, which 
will then take the form : 

(7) [m—x) {n—y) = m^n^A. 

These curves are the arcs of the hyperbolas which have for asymp- 
totes the sides of the rectangle of the extreme errors. At the limit 
these curves coincide with their asymptotes, and become the sides of 
the rectangle of extreme errors, that which is exactly true in the hy- 
pothesis of the two deviations, which is that of the curve of probability 
adopted, requires that we admit a maximum error, for all the points of 
the right line which bounds this rectangle have the same probability of 
being hit, which is zero, and without this rectangle the jjoints have no 
probability; when we approach the center the curve of equal probabil- 
ity takes an elliptical form, from which it departs more and more as it 
becomes rectangular. 

If in equation (7) we make x and y successively equal to zero, we will 
have the lengths of the semi-axes of the oval : 

x=7n(l—mnA) Sindy=n (1— mwA), 

, X m 

whence, - = — 

y n 

The axes of these ovals are in the same ratio as the extreme errors, 
or as the mean errors, which are themselves proportional to the square 
roots of the mean squares of the errors. 

General Didion, in speaking of the formula of Laplace, has shown 
that the curves of equal probability are ellipses of which the axes are 
proportional to the square root of the mean squares of the errors ; in 
fact, the curves nearly coincide, although they are different in their 
properties. But the deductions which we can make with reference to 
the real form of these curves are very uncertain ; their forms depend, 
finally, upon the successive inclinations of the tangent, or in other 
words upon the successive values of the derivative of an empirical func- 
tion, and it is very certain that two functions can be nearly coincident 
without at the same time having the same derivative. It does not sig- 


nify, then, if these derivatives should be coustantly very different ; on 
the contrary, it is more probable that their mean will be nearly the 
same, but still it is not less prqbable that at any given instant they 
assign to the tangents of the functions from which they are derived 
directions entirely distinct. The formula of Lax^lace, which in an actual 
case can only be considered as an approximation, is no exception to this 
rule ; we have here a remarkable proof of this, for, near the limit of the 
errors, the curve of equal probability should nearly coincide with a 
rectangle : in this respect, in spite of its less precision, the right line 
agrees better with the facts. 

** YIII. 


The probability of hitting within the rectangle or curve will be, as 
we have akeady shown, proportional to the volume standing on the 
rectangle, or the area inclosed by the curve, as a base, and bounded 
above by the surface of probability. 

It will be then generally of the form 

(8) P=B,^, 

in which B is the area of the base, and z the average height ; so that 
B^ represents the volume stauding on the base B. 

The value of z for any point of the base is given by (5) 

^ (m-a?)(n-y) 

from which, its average value for the required area can be found. 

In the case of a rectangle, if c and c\ d and (V represent the co-ordi- 
nates of the four corners, relative to the extreme errors, we easily 




If this rectangle is symmetrical to the axes of x and y, we wiU have 
for a rectangle of double dimensions, having its center at the mean 
point of imx3act, 

m'-}i- ' 

and calling a and h the half sides of this rectangle, 
^ /'2a a-\ r21) 'b-\ 

*TMs discussibn is considerably abbreviated from the one given in M, Breger's 
Memoir. C. A. S. 



Substituting for m and n^ Zy and Zy'^ we have the formula deduced 
by M. Helie in his Cours de Balistique : 

^^^ V3r vy w vv* 

These two last expressions can also be obtained by remarking that, 
in order to strike within a rectangle having its center at the mean point 
of imi)act and its sides parallel to the axes, it requires the concurrence 
of two simple events; first, to have an error in direction less than a; 
second, to have an error in range less than h. The probability required 
is then the product of the probabilities of these two events, of which 
the values are known (2) and (2)i. 

If the sides of the rectangles are proportional to the extreme errors, 

P will take a more simple form ; calling ^ the ratios — and -, we have 
(10) P=(2 Ic^^f^ 

and calling W the ratios - and —j, 

(10) ^ = {%lc'-iW^)\ 

The plans of powder magazines, store-houses, public buildings, &c., 
are generally of rectangular form ; and these expressions can be easily 
applied to them under the circumstances of elevated fire. The formula 
P=B^ will give easily the probability of hitting a circle, an elUpse, a 
rhombus, &c., these figures being situated entii'ely in the same right 
angle of the target, and having their axes in the required directions. 
Without enlarging upon these questions which have no need of any de- 
velopment, we will make, however, one remark which has some impor- 
tance ; we can, as we have seen, find the volumes intercepted between 
the rectangular hyperbolic paraboloid and the bases of different forms 
without the help of analysis by using the remarkable property* of the 
center ordinate, a property little known if one judges by the expressions 
for such volumes in the standard works. In using the right line as the 
curve of errors, we can, perhaps, render the stud}^ of the probability of 
fire accessible to a greater number of persons, at least in that which is 



The probability of hitting a circle of radius r, of which the center is 
at the mean point of impact, is obtained in finding the volume of the 
cylinder having this circle for its base and bounded above by the sur- 

* The remarkable property referred to is that the area of the base nmltiplied by the 
center ordinate gives the intercepted volume. C. A. S. 


face of probability 5 we will use for this formula (4). As tlie y must 
satisfy all points of the curve, the first integration will be between the 
limits zero and ^y^—x^\ that with reference to x will be between the 
limits zero and r j this will be the probability with reference to a quadrant 
of the circle situated in the first right angle of the target j for the whole 
circle we will have then : 

the integration is effected by ordinary methods, and we find : 




- 2r 2r 

2 3m 3n 

or, in terms of the mean errors : 



This expression has the considerable advantage of being applicable 
to the ordinary case in which the mean errors have different values. 
In those which have been determined thus far we have been obliged to 
suppose these two errors equal, on account of the difficulty of integra- 
tion. When this equality really exists we have, calling K the equal 


— and —J 
m n 



These formulas which* we have deduced, and those which we will de- 
duce, can be given a precision nearly absolute by means of a correction 
which we will determine hereafter ; but this correction is generally so 
small that in practice it is entirely unnecessary. 



The probability of hitting a right-angled triangle, such as OAB 
(Fig. 6), will be the volume of the cylinder having this triangle for its 















base. Calling OB and AB, h and h, we will have for all points of the 
right line O A, 2/=^^ j the probability which we require will then be 


which easily gives 

1 f*^ r~ 


^ ^ 2mn\ 3?^ Zm 4mnJ' 

In the case where the triangle considered extends as far as the side 
ED of the rectangle of extreme errors, as OOD does, we make 6=m, 
and we have 

(1^) • • • ^=6-»0-a)- 

For a triangle, such as OBF, of which the hypotenuse coincides with 
the diagonal of the rectangle of extreme errors, the ratios — and — will 
be equal, and calling them K, formula (12) becomes, after reduction, 

(14) . p^(?K-^! 

We have already seen that (10) the probability of hitting a rectangle of 
which the sides are 2h and 2h is 


we see easily from this that the probability of hitting the triangles, 
such as OFB and OFI, is the same, and that the whole target is divided 
into eight triangles, such as ODE, OEG, &c., which have the same 

probability, namely, g* 

^ We arrive at the same result by a different method. If we wish to 
find the probability of hitting the line IF parallel to the axis of x, and 
limited by the diagonal OE, or, rather, the rectilinear element IF%, we 
must integrate the expression which gives the probability of hitting a 
point (3) between the limits zero and ii?i=IF. Substituting the constant 
value ^i=FB for y we have, thus: 

dp'=^ ^^- ±^ay ; 

in the same manner the probability of hitting the element YBdx will be 

( m-xi)( nyi-^) 
df'=^ ^>^ ^^-^dx, 


we will have, consequently, 


#' («-,,)(m..-f) 
observing that 


we see that the denominator of the fraction ^S"i^ equal to its numer- 

ator, and that 

dp"=d;p' ) 

the two triangles O B F and O F I being composed of the same number of 
elements xdy and ydx^ having two and two the same probability of be- 
ing hit, it follows that 


or, P''=P'. 

This equality is applicable to all the right-angled triangles described 
upon the diagonal of the rectangle of the extreme eri'ors, even when 
their bases do not coincide with the axes of x and y^ and this does not 
result from the emi)loyment of the right line as the curve of i)robabili- 
ties. It is always true in the hypothesis of the two rectangular devia- 
tions, which should be the representative function of this probability. 
This function, indeed, contains always a constant which must be de- 
termined by experience ; this will be the square root of the mean of the 
squares of the errors ii^ the mean error ^, the probable error Y, the ex- 
treme error 3^/, &c.; in a word, a particular error c occupying a deter- 
minate position in the series of errors such that the variables x and y 

X If 

can be replaced by other variables —and -75 it is then clear that, if in 
two functions of the same form we can make equal to each other the 

X 11 

ratios -and-,, since they represent equal probabilities 5 it is this which 

takes place in the case we will consider hereafter. 

We see, therefore, that the probability of hitting the right triangles, 
such as OBF, OFI, FAH, becomes known the same as b}' the formula 
of Laplace; that is, the half of the probability of hitting the rectangle 
constructed on their hypotenuse as a diagonal. 

The probability of hitting the triangle OAL having its angle at the 
origin, the side opposite the angle parallel to oy will be the difference 
of the probabilities OAB and OLB ; in particular, that of hitting the tri- 
angle OKE, which extends to the extreme error DE will be, calling h 
and li' the heights of the triangles OGD and OKD ; 

(^^^ Q\ n 4.1V' )■ 


The different formulas contained in this paragraph are repeated in 
terms of the mean errors ;' and y'; the letter K' representing the ratios 

h b 

-7 and— when these ratios are equal: 

(13)1 OCD P= IsQ-wd' 

(14)i^ OFB P= g(^ ^ J. 

_ 1 fh-h' ¥-W^ 
- 18V r' 





The probability of hitting the triangles occupying with reference to y 
the same positions which these triangles occupy with reference to the 
axis of X is obtained by changing htob and y to /, 

i XI. 


To find the probability of hitting a trapezoid ABOD (Fig. 7), we 
call h the abscissa OE of the point of intersection of the right line A B 
with the axis of x, and h the ordinate OF of the point of intersection 
of the same line with the axis of 2/ j then the equation of the right line 
AB will be, 

if we call c and d the abscissas of the points A and B, the probability 
required will be 

1 r^ r-h+^ 


and performing the operations indicated: 


P = 





m Ti 

y^. l + id-oQnn-^e-M 

2 yj 

This formula is complicated, but from it we can deduce a number of 
others more simple j thus, in making d = b yve have the probability of 
hitting a triangle such as ACE j in making e = o. we have the proba- 
bility of hitting the trapezoid ODBF; multiplying by two and adding 
to it that of hitting the rectangle ABCD (Fig. 8), we have the proba- 



C o jt ~ 

Ji — !b ; 

bility of hitting the gable-end of a house in direct fire. If we make at 
the same time d = h, and c = o, we have that of hitting a triangle such 
as OFE (Fig. 7). In each case the formula is simplified more or less; 
in the last case, for example, we find for the triangle OFE, 




7i h Jih \ 

or in terms of y and y' 








^y' ' 108// 


for the rhombus formed by four symmetrical triangles situated in each 
right angle, we multiply by four the value given above j in the particular 
case where h—n and b—m^ that is to say, when the angles of the rhombus 
are at the middle points of the sides of the rectangle of extreme errors, 
we have for P the fixed value f j one has five chances out of six of hit- 
ting it, while the surface is only half that of the rectangle, which con- 
tains all the shots. The right line FE (Fig. 7) can take any position we 
wish ; we should only remark that li and b can become negative j thus 
AB and BGr (Fig. 10) being the sides of the rectangle of extreme 
errors, we will have the probability of hitting the triangle EFG in mak- 


ing <^=mand l)=c in the general formula (16), and in observing that 
h=OD is negative ; that of hitting the triangle HIJ by subtracting from 
the known probability of the rectangle OKU that of the trapezoid OHIK, 

. ^/_ ^1 / c ^ 

and we will obtain this with the same formula in which &=0T is nega- 

tive, and in which we will make e=o and h=-, — -^ j and in the same 

manner for the triangles or the trapezoids occupying the most diverse 
positions, thus there will be no dif&calty in calculating the probability 
of hitting any right-lined figure. But these remarks are particularly 
useful when the object extends beyond the rectangle containing all the 
shots, we then calculate similar probabilities, as, for instance, when we 
wish to hit the space comprised between the two right lines LM and NP. 

^ XII. 


The probability of hitting an ellipse whose equation is y=- y/a^—x^^ 

of which the center is at the mean point of impact, and in which a and 
& are the semi- diameters, is obtained by solving the equation 

4 /*« Pa 

which gives 

(18) . . . 

~ mn\2 3m 3n 4:mny'' 
or in employing the mean errors, 

2ah/^7z 2a 






We see making a=:h we have the expression already found (ll)i for 

the probability of hittin^^ a circle. When the ratios — and - are equaL 

m n 

we have, callins: them K. 


this, as we have already seen, is the probability of hitting a circle when 
iw=w and when K= — Thus the probabilitv of hittin^r an ellipse of 

which the axes are proportional to the mean errors, is equal to that of 
hitting a circle of which the radius is in the same ratio with the mean 
errors supposed equal in the two directions. This property is true 
whatever curve of probability is adopted, because we can show that 
in such an ellipse, first, the right lines F D and D E (Fig.ll), parallel to 

1 y" 









the axes, and intersecting the diagonal of the rectangle of extreme 

FT) T)F 
errors, are such that =^7-5 second, that these right lines, and con- 

r r 
sequently the right lines HF and EI. have the same probability of be- 
ing hit. It results from this that these ellipses are divided by the axes 
ox and 03/, and the diagonals of the rectangles of the extreme errors into 
eight sectors which have the same probability of being hit, whatever 
be the curve of probabilities employed ; we see thus that the probability 
of hitting within the ellipses of equal probability found by the formula 
of Laplace becomes known, since we know how to find that for a circle 
for which the probability is the same. 

We can find the probabihty of hitting these ellipses in a more direct 
manner. Placing the lormnlar of Laplace under the form 


l^e ^^ 

in which the constant v^ is the mean of the squares of the errors : the 
probability of having an error x will be 






that of having an error y will be 

and that of hitting any point, or rather an element dxdy^ will be 


We have the same probability of hitting each of the points of the 
curve of equal probability j particularly that which forms the intersec- 
tion of this curve with its major axis, for which x equals the semidiame- 
ter a and y equals zero, will have the probability of being hit: 

1 -^ 

This value can then be taken for all the points of the arc of the ellipse. 
The area A of the curve is rrahj and if we call h the constant ratio 

^, we will have A =V^, whence, taking a as s> variable, <?A= — ^^ — , 

which is the area comprised between two ellipses infinitely near, for 

which the ratio y is constant, we can admit that the probability d^ of 

hitting this particular elliptical element is equal to #P times as many 


elements dx dy as there are in this surface; that is to say, to ^"^^^I*? 

replacing d/^V and dA by their values, we find 

whence we have 

1 — ^ 
dV=—jje 2^*^ ada^ 
uu'k ' 

1 r^.^ 

uu'Jc J ' 


which, being directly integrated, gives 



and, since in the ellipses of equal probability we have -=— ^, it becomes, 

which is the probability of hitting a circle of radius a when the mean 
squares of the errors are equal, and consequently the mean errors. It is 
unnecessary to remark in this case that the circle is a curve of equal 


For the parabola of which the equation is y^=2px, we will have as the 
probability of hitting the portion situated between the curve the axis 
of a; and the right line x=b, 

P=^— -, I {m—x)dx I '^^^{n—y) dy. 
We easily find 

26. ^'' 

mn\3\p 6m 2n Smn J 

If we call Ji the ordinate corresponding to x=h: 

p_ hh /'2 2& h hh\ 
mn V3 5m 4in QmnJ' 

When &=m and ?i=7t, P has the fixed value ^) the parabola has then 
the equation 




Requiring the probability of hitting one of the curves of equal prob- 
ability which we have already determined, and which are formed by four 
equal and symmetrical arcs of hyperbolas, we know that these hyperbolas 
have the equation (7) 
(a) {m—x)[n--y)—mH^A] 

but referring them to two axes formed by the sides of the rectangle of 
extreme errors, they take the form 


From equation (a), the value of x when 2/==0, is x^m—mhiJ^, Letting 
c=m-?iA, we have for the probability of hitting within one quadrant 
of the oval, 

■^"^^SF / (^i--^) clxj ^n-^y) dy, 

Mf- I ( m—x ) dj)» 



^=2«< 2^-2+'^ '<*«•«> 
Bubstittiting the raloe of e, we have 

or for the probability of hittifig aoy where within the whole oval j 

P2= (1 — mVA^+ m^^A^ log. 7nhi'A'). 

KoTE.— This demonstration is different from that in M. Brdger^s 
Memoir, but the result obtained is the same, 0. A. S. 

If in this relation we make A=0, P becomes equal to unity, that is 
to say, to certainty ^ in fact, this supposition reduces the curves of equal 
probabilities to their asymptotes, which form, as we have seen, the sides 
of the rectangle of extreme errors, which contains all the shots j giving 

A the value — ? which corresponds to the case where these curves are 

reduced to a single point, the center of the target, we easily find P=0. 
The quantity A is seldom given, but we can write it as a function of 
known quantities; if, for example, the hyperbola is determined by its 
I)oints of intersection with the axes of the target Oy and Oil?, we will have, 
by making x 'dud y successively equal to z^ro, 

A= — ^ — and A^ f * 

whence we find 

m n m n 

If we call K the quantity mnAj the probability of hitting the target 
within the curves of equal probability will take the simple form 

P^1«-K2^K2 log. K^ 
+ XIT. 


To determine the probability of hitting a circle of radius r, the co- 
ordinates of the center of which are a and ft, such that all of the circle 
is in the same right angle of the target, we write from equation (8) 


in which B is equal to -r^, X to m—a^ and Y to n—h. Substituting 
these values, we have 

(i») • p=^<'»-«) ('»-'')' 

or in terms of the mean errors 

^19)i ^=-8i5p ^^'^-"^ (^''■-^)- 

If the center of the circle is on the diagonal of the rectangle of ex- 
treme errors, the ratios — and - will be equal, and, calling them K, we 
m n ^770 7 


P=— (1-2K+K^). 


In the case where the mean errors are equal, and the circle tangent 
to the four sides of the square formed by tlie extreme errors and the 

axis of X and y^ we find for P the fixed value ^^ and for the four sym- 
metrical circles, ^ . 
These formulas can also be obtained by the solution of the expression 

P=^-^2 / ^« / (^>i-/>cosa-a) (7i-/?sina-&)/>^^, ' 

which we obtain by moving the axes to the center of the circle to be 
hit, and using polar co-ordinates. This requires very simple calculation, 
and often has the advantage of being useful when the circle is not all 
in the same quadrant of the target. Jf the center is on the axis of y we 
make a=0; the integration with respect to p is between the limits zero 
and r. Eemarking, then, that the probabilities of hitting the half circles 
situated on each side of the axis of y are equal, we make the second in- 
tegration between the limits — ^ and -j-^, when multiplying by two, we 


and if the center is on the axis of x, 

(3;:-4r) { m-^a)r^ 

Major Moncrieff has proposed replacing casemated batteries by cir- 
cular pits, lined with masonry, much larger at the bottom than at the 
top, in which are mounted guns upon carriages of his invention, which 


recoil below the surface. The pits are connected together by under- 
ground passages. This system appears to be in great favor in England, 
where it has been tried. Among the methods proposed by the artiller- 
ists of this nation, we find the arrangement of the pits in groups of 
lives. Let us find the probability of hitting one of these groups of fives, 
considering as good only those shots which penetrate into a pit, sup- 
posing that we aim at the center of a group. 

Let us take the diameter of the pits as 6 meters, and suppose that 
the center of the central pit is the mean point of impact, and the origin. 
Let the co-ordinates of the centers of the other four pits be ±5 meters, 
so that they will be arranged symmetrically with reference to the origin. 
Formula (11) gives immediately the probability of hitting the central 
pit, and (19) that of hitting any of the others 5 letting r=3, a=b=^, 
and we have 

^^-y/ 3r/' 3fr' 2fr'^ 

for the central pit, and 

_ rr __ 5- __ Stt 257r 

J^a = — , 3-/2 3PP + 9PP 

for each of the other four pits. Therefore, the probability of hitting 
the battery, or of having a projectile enter a pit, will be the sum 
Po-f 4Pa, which we can write, taking 7:= 3. 14, 

V'-^) "— ^^/ 2 / /2 "I" 9 /2' 

m7 22.3 22.3 . 35.4 


We should call attention to the fact that this formula is only ap- 
plicable when the five pits are contained within the rectangle of the 
extreme errors, which requires that 7' and y' should both be greater 
than 2"^. 66 ; if, for example, the mean lateral error is 3 meters and the 
error in range 20 meters, we find that P=0.206; we would have 21 
chances out of 100 in our favor, and, if we fire a great number of shots 
one-fifth of them would enter the pits. We find the probability of hitting 
the central pit is 0.048, and that of hitting each of the others is 0.041. 
In consequence of the greatness of the errors, these two probabilities 
differ but little. It results evidently from this, and whenever the mean 
errors are large it would be better to aim at the center of the battery 
than at each of the pits in succession, and if a number of pieces are 
firing we would not assign to each of them a particular object to be 
aimed at, but would aim them all at the central pit. There is also 
another result to be deduced from this, which is that it would not 
require many more shots to silence the whole battery than would be 
required to silence a single isolated pit, for the most probable case is 
that the projectiles that hit would be divided amongst the pits propor- 
tionally to their respective probabilities, which at the same time agrees 
with good sense and the theory. It is true that the probability of this 


division being perfectly exact is very small: but it will differ bur little 
from This, and will rapidly approach it as the nuuiber of shots increase. 
Therefore, we will have a very great probability that the number of 
shots which will hit each of the pits will be at least ecjual to a number 
slightly less than that assigned by the most probable division. If. for 
example, we tire lUU shots, under the above conditions, it is logical 
to admit that the central pit will receive live shell and the others four; 
but there is a very great probability that each pit will receive at least 



In order to find the probability of hitting the deck of a vessel moving 
in an oblicjue direction with regard to the plane of fire, we should take 
account of the exact form of the deck : but we coasider that this pre- 
cision is useless in practice, and that it is sufiicient to consider a 
rectangle of which the dimensions are a little less. T\'e will take for 
the length that which is called the lengfJi leticeen pcr]jendlculars. and 
for the width the beam at the waterdine. information which we can 
easily procure from maritime ])ublications. Simplified thus, the prob- 
lem, like must of the preceding, can be solved with difiiculty by the 
ordinary methods. TTe can only, by other means, divide the rectangle 
to be hit into a series of small rectangles, of which the sides are parallel 
to the direction ot the axes, and calculate separately the probability of 
hitting each of these, an operation impracticable unless we have a great 
deal of time at our disposal ; even then we would neglect a series of 
triangles, and. in order to have a good approximation, it would be 
necessary to carry the subdivision of the rectangles very far. In this 
way We would lose the benefit of the precision which ought to follow 
the use of the regular formulas. 

The ciuestion is. on the other hand, easily solved by the methods of 
this article : we have, in all eases, to find the probability for trapezoids 
and txiaiigles. which, after what precedes, offers no difiiculty. It is 
evident that in consecjuence of the relative size of the ship and the 
magnitude of the mean errors the figure to be hit can take very different 
forms, for the precision of the pieces of the new artillery makes the 
object extend beyond the rectangle of the extreme errors, which con- 
tains all the shots, and consec[uently gives different forms to the portion 
of the object to be considered. However, if in real fire we use directly 
the mean errors given by the tables of fire we may make a great error; 
these quantities may vary considerably in consequence of errors not 
inherent in the piece, TTe conclude from this that the object to be hit, 
if large, is more fi'equently than we wuuld believe contained entirely in 
the rectangle of the extreme errors : thus. also, when we wish to use 


elevated fire by diminishing the charges, we will find that the errors in 
direction are notably larger. However, the mean error in range is al- 
ways much greater than the mean error in direction, consequently on a 
horizontal target the rectangle of extreme errors has a very elongated 
form. It results from this, that in practice three cases present them- 
selves, which we will examine successively. 

In the first case the mean error in direction is very small ; the proba- 
bility of hitting the ship IJ is then reduced to hitting the parallelogram 
EFGH, and one obtains it by multiplying by two the sum of the proba- 
bilities of the trapezoids OKGP and OKFL. Continuing to call for the 
present /t and h the distances OK and OS, the probability of hitting 
the trapezoid OKGP will be given by the general formula (16) XI, in 
which h will be negative, d=m and c=o; the same formula will give the 
probability of LFKO, but in supposing this trapezoid placed in the first 
quadrant we see that h has the same value, that b becomes positive, while 
d=m and c=o. Consequently when we take the sum of the two ex- 
pressions, all the terms which contain b to the first power, having con- 
trary signs cancel each other, those containing c become zero, and we 
can directly write for the probability of hitting the ship 

If now we call 21 the width of the vessel and w the angle which its axis 

makes with that of x, we have b. 

sm (o 

and h: 


J these values 

substituted in that of P give 

P=l( 2» 

m^tan^ «> 

W V cos <o cos* iii 




and iu terms of the mean errors; 



_i r ^r' 

3/\cos io 3 cos^ 




When the ship makes an angle of 45° with the line of fire, this formula 

and if (o equals 60°, 






3/=^V^ 3 21/ 

Finally, if w is equal to 0^ or to 90^, the rectangle to be hit will have its 
sides i)arallel to the directions of the errors, and we can make use of 
formulas already found ; in this last case it will be useful to know the 
length of the ship. 

Most of the ironclads of the squadron of all powers have at the 
water-line a width but little differing from 17 meters. For precision , 
we will take the Eichelieu, to which this width applies, except the deci- 
mals, wUich we will neglect, and of which the length between perpen- 
diculars is about 98 meters ; if the fire takes place from a distance of 
1,041 meters, with the rifled shell of 22 cent., charge 1 kil., conditions 
under which the mean lateral error from the tables is 1.90™, and that of 
the range 27"^.9, we find the following results : 

for a>=0o 450 6O0 90O 
P=0.17 0.27 0.50 0.84. 

We see that we would greatly deceive ourselves if, when a ship makes 
an angle of 45^ with the line of fire, we take for the probability of hit- 
ting it the mean of the extreme probabilities. The preceding formulas 
are applicable to all cases in which I is equal to or greater than ^y sin w ; 
if I is less than 3^ sin oi, the point P (Fig. 16) falls without the ship, as 
in Fig. 17, at D. We obtain the probability required by subtracting' 
from that for OABD the j^robability for FOD, and adding to the dif- 
ference the probability for E A, and multiplying the result by two. 


For tbis we can use the general formula for a trapezoid, and, calling 
li and h the lengths AG and OF, we find for the trapezoid OABD, 


n — 

mil mn 




the triangle FOD, 



' m~ 


^ + 

mil mn 
36 + 3&" 





_ /i /wft^ 6/fc 7^& &2/i ^ 2m7i\ 
i>i-i?2-^^^2|^3— 2-3,^- ^ + 12l?^^+ ^^ 3F / 

For the probability of the triangle AGE we have 

h f 

n¥ hh nl) Ifli \ 
'3m' 3m"^m"^12mV' 

and calling P=2 {pi—P2+P3)i the probability of hitting the ship will be 

^ ^ 3n\4.m' m^ h hj' 

This, in terms of the mean errors, of the width 21 of the ship, and of 
the angle <y, gives 

,oox p_ ^ r l_ ? ^r' cos ^ 3r sin c.-\ 

^""^^ • 27r'^ cos^ a> V36^2 sin^ «> 3;- sin ^"^ I I J' 

When a>=45^ we will have 

_ 4^ /-J^ ^/2 _v_ ir_\ 
^'-27/^18^2" 3r '^l^/2~l^/2J' 

and when it is equal to 60^, 

4Z^ ^ 2^^ 4? V 3r 73 \ 
/' ^^-27^27^* 3rV3^ r ^ / 

Thus, the fire being against a ship of the same type as the Eichelieu, 
and generally against an iron^clad of 17 meters width, but with the gun 
of 16 cent, of model 1864, at the distance of 3469 meters, for which the 
mean errors in direction and range are 4".9 and 30 meters, we will have, 
according to the value of w : 

forw=Oo 450 6O0 900 

P=0.18 0.24 0.32 0.69. 

We can still use these formulas when 3^ becomes greater than L cos 
«y+7 sin w^ although, strictly, we would only have the right to use them 


up to 3;'=L cos (o — I sin w, but the error which can result from this 
extension is so small in practice that it will not be of consequence. 
Beyond this, the rectangle replacing the deck of the ship will be entirely 
comprised in that of the extreme errors. We are then led to a formula 
less simple than the preceding ones, whose simplicity is very satisfac- 
tory, considering the nature of the question. We can still employ the 
division into trapezoids and triangles, but we arrive more easily at a 
result by taking as the axes of x and y that of the ship and a perpen- 
dicular to it. For greater simplicity we will consider the axis of the 
ship at an angle of 45^ with the line of fire; this will give between the 
former co-ordinates and the new ones the relations : 

x=(^-Y) Vf and 2/=(X-f Y) Vj: 


Substituting these values in the equation of the surface of probability, 
and finding then the probability of hitting the rectangles FDBE and 
AIHG, we have 

Pi=^?^. j^^xJlm-(X-T) ^/2] [m-(X+Y) V2]dT, 

to which we must add four times the probability of DOF, giving 
^ ' ^ mn L 3 \m n J Qmnjy 


a comi)licated result, but in applying it to a single type of vessels it is 
reduced to a formula of four terms, with numerical coefficients. This 
reduction we will not make here; we will only consider the general 
methods which can be simplified in the works on the study of any i)ar- 
ticular kind of fire. We will give another solution of this same ques- 
tion; it is less exact as a demonstration, but it gives practically the 
same value; moreover, it is much more simple. 

Generally iron-clads haA^e a length greater than five times their 
breadth. In considering them as formed of five squares we neglect 
only the extremities, the probability of hitting Avhich is very small. 
These squares, except the one which has its center at the origin, are in 
the conditions indicated in YIII ; that is to say, they are arranged in 
groups of four elements symmetrical to two axes through their center and 
parallel to those of the target. We can then apply the relation P=BZ, 


or, what is the same thing, P=— ^-^, the co-ordinates of the points 

M and N relative to the extreme errors are 

Xi=m— Z-v/2; X2=m— 2i-/2, 

Yl=7^-^^/2'j Y2=n-2ZV2, 

and for each of the squares B=4?2, we find thus, for the probability ot 
hitting the four squares M, N, P, and Q : 

472 _ 

Pi=,,^ [4:mn-6lV2{m-{-n)-{-20l'], 

We know the probability Pg of hitting the square (a), and we finally 

^ 41' r 19lV2^ 12W1 

(23) ^=m%i^'^''— 3~(''^+''^+T"J' 

or in terms of the mean errors, 

472 p _ 121 Z^ 1 

(23), ....... P=gjpp[45;./_19?V2(x+/)+--g- J. 

Nothing prevents the employment of the same method for ships in 
which L = 6Z ; it is sufficient to add to P the probability of hitting two 
squares s, and two squares r of which the sides are Z, we obtain a for- 
mula similar to the above ; the numerical coefficients alone have differ- 
ent values, and it is evident that it would also be similar if we substitute 

for the ship a rectangle such that j is any whole number, which will 

always be possible by neglecting small portions at the extremities for 
which the probability is insignificant. 

The formula (23)i applied to fire at the Eichelieu, with the piece of 16 
cent, from a distance of 6,848 meters, for which the mean errors in 


direction and range are l-j-'^.T and 42 meters, gives for the probability 
of MttLDg this ship when it is at an angle of 45-, O.IG; we easily find 
the values of P for a; = () and cv==90-, and we have : 

cu=^(P, 45-, 90^, 
P=:0.13. 0.16. 0.25. 

In comparing tliese results with those obtained at distances of 3469 
and 1041 meters, we see that the probability increases more rapidly 
when w approaches 90^ : that is. when the ship is perpendicular to the 
line of fire we have the least chance of hittiog its deck. A.t 45" the 
chances are a little augmented: it is beyond this that the increase be- 
comes notable : at 60" the probability is a little below the mean of the 
extreme probabilities, and at 90-- it has its maximum value. TVhen the 
mean error in direction is very small, the superiority of fire at 90- is 
enormous : i3Ut this superiority, although always sensible, diminishes 
as the mean error increases, or, for the same piece, diminishes with the 
distance. If, for example, the fire is from a battery^, we see the consid- 
erable advantage that there would be in taking advantage of the time 
when the ship is end on to the battery : under these conditionxS we have 
greatly increased chances of hitting at 6S00 meters with the piece of 
IG cent, over those of hitting with the same piece at 3500 meters if the 
ship is parallel to the battery. 

If the fire is from one ship against another ship, there would be more 
chance ofhittiug the deck of the adversary in chase or in retreat: how= 
ever, we give him the same advantage : but if the two ships occupy the 
positions represented below, and they have the same number of gnns, 


C CD will have an enormous advantage over his ad= 
versary. in supposing that the armor cannot be 
penetrated by shots from a distance for which shell 

D fije is still verv efi'ectual. 

There is some interest in knowing the amount of the error we commit 
in neglecting, as we have done in the last case, a part of the length of 
the ship. If. remaining under the same conditions of fire, we determine 
the probability of hitting the parallelogram ahcd (Fig. 18), formed by 
the sides of the rectangle to be hit prolonged to the extreme errors, it 
will be greater than that which we have found by employing formula 
(23) by a quantity less than 0.01. A ship of the same width and more 
than 110 meters in length will be entirely comprised in this figure. 

This will evideutly not be the same if the ship is in the line of fire, 
but in this case we take account of the actual length of the ship. In 
other respects the use of the right line as the curve of errors introduces 
an error in the calculation ; but this correction, evaluated as it will be 
hereafter, gives, for all the cases which we have just examined, only 
inconsiderable quantities, of which it would be puerile to take account 
in practice. 


-4— XYI. 


We could continue indefinitely the applications of this method, their 
number is unlimited; but the examples already given are sufficient to 
enable us to comprehend it, and we should be able to calculate without 
difficulty the probability of hitting an object of any form with an ap- 
proximation very near to the value which we would obtain from the 
ordinary formula, supposing that it would enable us to solve the ques- 

The certainty that we can easily find these values increases consider- 
ably the field of useful information that the calculus of probabilities is 
in a condition to furnish to practice; many questions which have been 
solved satisfactorily for other purposes become applicable to firing. 
There will be some modifications to be introduced on account of the 
nature of the problem. We will give one example to show the nature of 
these modifications. 

Often we know how many times it is necessary to strike an object 
with a given proj ectile in order to produce a certain result, which may 
be the ruin of an epaulement, the opening of a practicable breach, the 
destruction of a bomb-x)roof, the demoralization of a body of infantry or 
cavalry, &c. The experiences of the Island of Aix, of Graudentz, of 
Gavre, of Metz, <&:c., and those of former wars, furnish much informa- 
tion on this subject which it is easy to procure. Knowing, then, that it 
requires K projectiles which hit to produce the desired result, we can 
easily calculate how many it will be necessary to use in all, which is, as 
is evident, a question of the supply of munitions. The probability of 
hitting the object being j;, that of not hitting will be 1— jp or q; conse- 
quently if we fire n shots in succession, each of the terms of the develop- 
ment of (g'+i^)" will represent the probability of hitting zero times, 

once, twice, n times. Thus the probability of hitting at least 

once, at least twice, at least K times, will be unity less the first 

term, less the two first terms, less the K first terms ; in particular, that 
of hitting the object in n times will be that of hitting at least once, and 
we will have 

W p=i~r5 . 

and if we wish to know the number of projectiles which it is necessary 
to use in order to have a probability P of attaining the desired result, 
we must solve this equation for Uj which will give 


(c) w=,-''-), L 

^ ' log l~i?) 


The abbreviation log. means ordinary logarithms. If, for example, 
the fire is against one of the Moncrieff pits, of which we have spoken in 
XI Y, and under the conditions of the numerical example which it con- 
tains, conditions which assign to P the value 0.018, or simply O.Oo, we 
find the following results in making K=l ] that is to say, in admitting that 
a single shell, falling in the midst of the gun's crew crowded in the pit, 
and projecting its fragments upon a carriage as complicated as a disap- 
pearing one, would be sufficient to silence its fire. 

First. If one can use 30 shell, the probability of success will be 0.79, 
formula [h) ; 79 times out of 100 we will succeed in reducing the adver- 
sary to silence, or about four times out of five. 

Secondly. If we wish to know how many projectiles it wouid be neces- 
sary to fire in order to have the probability J of succeeding, we make 
P=J in the formula (c), and find ??=ll5 that is to say, if we only have 
14 projectiles we will have equal chances of succeeding and failing. 

Thirdly. If we wish to consider the certainty of succeeding, we give 
P a large value, for example 0.92 ; the same formula shows that we 
would have to fire 50 shell under the conditions already given, in which 
we take ,-=3'^ and ^'=20'^. 

When K equals iico^ or when it will require two shell to attain the re- 
quired result, we have, by subtracting from unity the last two terms of 
the development of {p+g)" : 

{d) p=.l_(,i^g'-i + gn), 

and in the same manner for K equals 3, 4, &q. ; yet we can only deduce 
the value of n by a series of trials easily made. Applying formula (d) 
to the same case, if we wish to put two projectiles into a Moncrieff pit, 
we find the following results : 
With a number of projectiles equal to 

100 50 30 

we have the probabilities of success, 

/ 0.96 0.63 0.45. 

r These methods are exact; but, when K is large the calculations are 
too difficult to be made directly. We can obviate this difficulty : the 
larger term of the development of (g+i^)'' represents evidently the prob- 
ability of the most probable case. K the exponent of 2? in this term is Z, 
the most probable case is when we hit the object I times out of n shots; 
7 is, in other words, the greatest whole number contained in p (^? + l), 
and \fn were a little greater, we would have I =pn, nearly. The well- 
known formula, 

2 / V2pqn 

can be considered as the sum of the terms (2?+^)^ of which the expo= 
nents of 2^ are comi)rised between l-^g and l-^-g. But if we make K 


equal to l—g, the problem will be the determination from this formula 
of the probability of hitting the object a number of times comprised 
between l—g and n inclusive. The integral above, having its elements 
positive and negative equal two and two in absolute value, we can write 

-vVy « 

■1/2 pqn 

1/2 pqn 

We conclude that the probability of hitting the object a number of 
times comi)rised between I and l-~-g is equal to that of hitting it a num- 
ber of times comprised between I and l-\-gj and consequently the proba- 
bility of either is 

V'2 pqn 

But the probability of hitting the object at least K times is composed, 
first, of the probability of hitting it a number of times comprised be- 
tween I and l—g, and, second, a number of times comprised between I 
and w, which we can without inconvenience make equal to infinity 5 we 
can then write, 

"i/ 2 pqn 


Whence we write, 

,., .... -i[i.^/?„].i±-^, 

in order that we may m ake use of the tabulated values of the integral which 
is contained in this formula, the calculation will then be extremely sim- 
ple. It is evident that ji)?*— K may be negative, and in that case we 
must subtract the numerical value of F (t) instead of adding it. This 
formula supposes that n is large, the degree of the approximation de- 
pends, then, upon the value ofp) if this quantity is very nearly zero it 
will give values of P too small, or it will require that n should be very 
large ; the contrary will take jjlace if ^ is nearly unity. This expression 
has a direct use since it enables an officer who has at his disposition n 
projectiles to see whether he is in a condition to obtain the desired re- 
sult ) we can use it otherwise for the solution of the problem which we 
proposed at first, that is to say, to find the number of projectiles which 
we must fire in order to hit the object at least K times with certainty 


or with a probability, P. We must find in the table of F {t) the value 
of t which makes this integral equal to 2P— 1^ and, in placing, 

V2pqn ' 

we deduce n) but ordinarily we know the two values of n which corre- 
spond to P =i and P^l ; in the first case F {t) must be zero, and con- 
sequently from its upper limit equal to zero, we have 

(25) NJ=|; 

and if we have only »i projectiles at our disposition we will have as many 
chances of success as of failure. In the second case we must remark 
that P can only become equal to unity for ?i=co ; and, indeed, when 
there is the least chance of an event we cannot have mathematical cer- 
tainty of its accomplishment, but it is evident that we can make this 
adverse chance less than any given quantity by increasing the number of 
trials. Practically, we can admit that when we have eleven chances out 
of twelve of success the result should be considered as certain ; we choose 
this value of P, or, better, P=0. 92135, because it makes F {t) equal to 
0.84270. We can then be assured that t is equal to unitj^ ; we have, 
(25), .... .^^^^g+K+V(g + Kf-i>-K-^ 

a formula which no longer requires the aid of a table; if ^ is sufficiently 
small, which often is the case in practice, this is reduced approximately 


(25)2 n^i= 


Expressions (24) and (25), and (b) and (c) give the same information, 
and are applicable to all values of K ; but we should not forget that 
they are only approximations, and whenever it is possible we should 
make use of exact formulas. 



When new causes of error independent of the piece come to be added 
to those for which have been calculated the mean error of the tables of 
fire, there results the usual principle of the theory of errors that the 
total mean error is the square root of the sum of the squares of the mean 
errors which each of these causes would jjroduce if it acted alone; if, 
5782 4 


for example, these causes act laterally, and they produce individually 
the mean errors 61, e^^ &c., the total mean error E will be 

if they act in the direction of the range, we will have in the same way 

Thus when we require the probability of hitting any object, by means 
of the preceding formulas, we should, in reality, put in place of y and 
y' the quantities E and E^ This supposes, however, that we can take 
for the curve of probability of these new errors that which is repre- 
sented by the formula of Laplace, and this will be the general case ; but 
this supposition is only formal; it is evident that the quantities e^ 
62 . . ei' e^'^ &c., should be determined by experience. 

To fix the ideas we will take a numerical example. We fire from a 
battery with the tubed rifle of 24 cent, against an iron-clad of the same 
type as the Eichelieu, the projectile being the elongated shell of 120 
Ml., with a charge of 2d> kil. The distance is 4,000 meters, for which 
the Memorial de VArtillerie de la Marine assigns a mean error in direc- 
tion of 1™.47, and in rangel9™.6 ; we have also the time of flight, 11^3. 
If there is no other cause of error, the probability of hitting this object 
when its axis is in the plane of fire will be 0.95, which we may call 
certainty ; but now we will take the actual conditions of the fight. In 
the first place, the distance is not known, and we must measure it. 
Suppose that the instrument employed gives, at 4,000 meters, a mean 
error of 150 meters, an amount that is not exaggerated if we consider 
that the measurement must be made as quickly as possible, the object 
being in motion. The ship is approaching or receding from the battery 
with a certain velocity, which we try to estimate in order to correct 
the sight ; but, in view of the few resources we have at our disposition 
to effect this object, a sensible error may be made. Admitting that 
the method employed permits a mean error of one knot, by which 
we assign to the ship a velocity which will be too great or too small by 
0™.51 a second, and, as the shell is in the air 11^3, there will result a 
mean longitudinal error of 5°^.8, consequently the total mean error in 

range E' will be 




In the total mean error in direction it will be necessary to take 
account of the mean error committed in the correction for the wind. 
Suppose that the very coarse method which we are obliged to use in 
our batteries permits a mean error of 2™, we know, from the work of 
M. Helie upon this subject, when the wind is across the line of fire, the 
deviation d which it will produce can be represented by the following 
formula : 



in which 

W is the velocity of the wind in meters ; 

T, the flight of the projectile in seconds } 

X, the range in meters j 

Y, the initial velocity in meters, which in this case is about 474™ ; 

a the angle of projection, equal in this actual case to 7° 44' 3C. 

In making W=2°^, we find cZ=4™.3, and this will be the mean lateral 
error which results from the imperfection of the method emplo^^ed of 
measuring the velocity of the wind ; if this is the only cause of error, 
we have 

Ei=/\/l.47+ 4.3=4^.5.* 

We have neglected the other causes of error heretofore, but now we 
will calculate the probability of hitting a rectangle of 98™ in length by 
17™ in width, when the total mean errors are 151.8™ and 4.5™ ; and we 
find P= 0.18. Thus, in spite of these corrections, this probability, and 
consequently the useful effect, will be at least five times less than on a 
range where we can eliminate these causes of error ; it would require 
in general five times more projectiles, and time, to jiroduce the same 
result, and the piece only performs a small part of what it could be made 
to do. 

We can also verify the fact that the error in the distance alone reduces 
the probability to 0.20 j that caused by the error in the estimation of 
the velocity of the wind reduces it to 0.82. We see that a simple error 
in a correction which is sometimes entirely neglected produces an appre- 
ciable eflect, since it causes the loss of about ^ the useful effect. It is 
of interest to see what result we would arrive at if we neglect this cor- 
rection : the velocity of the wind being 10 meters a second, the formula 
of M. Helie shows that there will result from it a lateral deviation of 
21™ J but this we would not consider a variable error, of which we can 
determine a mean to which the law of ordinary probability is applicable; 
it is evidently an accidental error, and it produces a deviation of 21™ to 
the right or left according to the direction of the wind. The mean 
errors do not change their values, consequently the probability sought 

*We find in the Lessons on Practical Mechanics, by General Morin, a table of the 
pressures of the wind upon an immovable plane surface of one square meter, result- 
ing from the experiments of the English physicist Rouse : for velocities of 8 and 10 
meters the pressures are 7^.4 and ISi^..^, and their difference is GKl, consequently, if we 
make an error of two meters in the velocity of the wind in the case now before us, 
the error would be the same as one of G^^.l in a pressure on a surface of one square 
meter. Admitting, first, that upon a convex surface, like that of a projectile, the 
action of the wind would be the same as upon a plane surface equal to its section on 
the axis; and, second, that on account of the small velocity given to the projectile by 
the wind, its pressure will be constant, we can calculate approximately the deviation 

required : this will be the space j)assed over by a body whose mass is , and sub- 
jected to an effort of 6^.1 during the time t, which in this case is 11^3. This gives 
d = 4™. 


is that of hitting a rectangle having its sides parallel to the direction 
of the errors, but whose center instead of being at the mean point of 
impact has for its co-ordinates 21™ and ; the rectangle to be hit will 
then be completely without that which contains all the shots, and the 
probability will be equal to zero. The ordinary formulas give a value 
expressed in decimals so small as not to have any practical value ; con- 
sequently this neglect is sufficient to reduce the probability from almost 
certainty, 0.95, to the certainty of failure, 0. There results from this a 
curious fact, that with a piece less precise, or if there exist other causes 
which increase the total mean lateral error, we would have more chances^ 
of hitting the object, since it would be in all, or in part, contained in 
the rectangle of extreme errors. Under certain circumstances these 
chances can have a sensible value ; thus, in the above case, supposing 
that the mean error in range remains tbe same, but that on account of 
some cause the total mean lateral error becomes 10 meters, we find that 
the probability of hitting the deck of the given ship, when we neglect 
taking into account the wind with a velocity of 10 meters, is 0.17 (8) 
YlII j that is, we hit about once in six times. We must guard against 
concluding from this, as is sometimes done, that in certain cases there is 
,in advantage in the employment of pieces less perfect that those of mod- 
trn artillery ; this proves simply that it is indispensable to apply the 
correction for the velocity of the wind, for it is very evident that, coarse 
though the means employed may be, yet they enable us to estimate this 
velocity with sufficient exactness to give a probability far higher than 
0.17, and that the advantage of precision in the piece becomes thus evi- 
dent. Thus, suppose that the mean error in this example, due to the 
evaluation of the velocity of the wind, is doubled, that is to say, that we 
make an error of 4™ in the velocity of the wind, more or less, an error we 
would hardly make if we remembered only two or three of the values given 
in the special tables for different forces of wind ; for instance, in the Aide- 
Memoire cfArtillerie de terre, we find that the probability would be less- 
ened by this error of 4™ to 0.54,* that is to say, we would have three times 

* We must here define the meaning of this mean error of 4™, which we have attrib- 
uted to the judgment of the observer. Suppose that a person having some ideas upon 
the velocity of the wind makes a very great number of estimations, and that, without 
his knowledge, for' otherwise his judgment would correct itself little by little, we 
note each time the mean velocity by an anemometer sufficiently correct; taking, 
then, all the trials where this velocity was lO'", or where it did not differ from this 
quantity by an amount sufficiently large to be considered, we would have the mean 
of the absolute errors committed. It is to this mean that we attribute the numerical 
value of 4™. Introducing it as a constant in the expression 7/ = ce— " < , we assume 
what is not perhaps completely justified. It would be necessary that the mean of the 
estimations should be about 10™, that is to say, that there should be an almost 
perfect equality between the sum of the positive and negative errors ; but this only 
experience can show. Not having any data for the question, we are obliged to em- 
I)loy this method, of which the exactness can be considered as probable, but not as 
certain ; nevertheless, we can evidently regard the result to which it leads as a rough 
approximation, and this will be sufficient for our imrpose. 


more chances of hitting than in the preceding case, which reduces to 
nothing the paradoxical assertion with which we commenced. We 
should, moreover, remark that, in order to obtain the probability 0.17 
without taking into account the wind, we must suppose an imaginary 
piece, like the gun of 24 cent., of which the mean error in direction is 
seven times greater, without the mean error in range being changed j 
but, in reality, this last would also have a value much greater, which 
would sensibly diminish the chances of hitting. We can easily see that 
with the shell-gun of 22 cent., rifled and hooped, which at 4,000 meters 
has a mean lateral error of about lO'", the chances of hitting will be 
almost nothing, if we do not take account of the wind. Hence if we 
tire with equal care two pieces of unequal precision, it is evident that 
the superiority will always be with the one whose mean errors are the 
smallest ; but the calculation gives the numerical value of this supe- 
rioritj' , and shows without difficulty that it is weaker as the corrections 
are less precise, so that, when the causes of error independent of the 
piece are very great in comparison with the mean errors, the chances 
of hitting become little different whichever piece we employ. We must 
not conceal from ourselves that the want of means of correction in 
which we are actually i3laces us nearly in this condition. 

When, instead of being in the i^lane of fire, the ship is j)erpendicular 
to it, we arrive at analogous results ; we remark that the velocity of 
the object will have in this case as marked an influence as that of the 
wind in the preceding discussion. 

We are struck with the enormous loss of useful effect in the fire of 
the pieces of the new artillery from a distance when we only have such 
means as we use at present for correcting them. The instrument is 
one of great precision, but the imperfect manner in which it is used 
renders this precision almost useless. The conclusion which we must 
draw from this is that we must reduce as much as possible the errors 
which we can commit in the instantaneous estimation of the distance, 
and also of the velocities of the object and of the wind. There is no 
question more urgent in requiring a solution than this, and we will 
admit without hesitation that, no matter how costly these means of 
estimation may be, at whatever price the instruments that would 
accomplish this purpose could be purchased and placed in our batteries, 
there would be a great economy, for nothing is more ruinous than the 
waste of our large modern projectiles. The figures at which we arrive 
in this paragraph show also the grave responsibility which rests on the 
officer who directs the fire j without doubt his talents and his experi- 
ence have a considerable influence upon the value of these corrections, 
which are of sufficient importance to change the certainty of hitting the 
object into the certainty of not hitting it. These reflections are particu 
arly applicable to fire at sea, since to the same causes of error are 
added that which we may make in our own speed ; they show, in both 
cases, what would be the true nature of a fight at a distance under these 




Before finishing this memoir we will make some useful developments 
of the nature of the error which is introduced in the calculation by the 
substitution of the right line for the ordinary curve of errors, as is 
shown in this figure. It is of interest to compare the ordinates of the 


curve and those of the right line, which, multiplied by dx^ represent 
the probability infinitely small of having exactly the error x. We will 
then compare the successive values of the probability of not exceeding 
the error x^ or the areas between the axis of s, that of x^ a variable 
ordinate, and either the curve ABO or the right line DE, the former 
being given by the ordinary formula, which, in terms of the mean error, 
can be put under the form : 





and the latter, as we have seen (2), I, by 

p _2x x^ _2x x^ 

If we compare the successive values of P and Pi for the same values of 
-, the greatest difference is about 0.02, which corresponds to - equal to 

* In tlie note of Article III it was slioTrn that tlie mean error y was equal to 

1 XT, „^„„„ 7 1 ^ 7„ ic 

therefore h- 


the usual form. 


and lix: 


Making t=zhx, the above expression takes 


1.35 and 2.70. The mean difference is 0.0125, which can be entirely neg- 
lected in practice, when we consider that the tables of fire from which 
we take the mean error have been compiled from a comparatively few 
rounds at each range, and that consequently the mean error is not de- 
termined with a degree of precision at all comparable with the above 
difference. If, however, we simply wish to determine the probability 
of not exceeding a given error, we would make use of the ordinary 
formula, for the value of which tables have been calculated j the em- 
ployment of the right line would in fact be useless if in practice we 
were only led to such simx)le questions. That we may know i)erfectly 
the nature of the approximation which we use, we will make a few more 
remarks which have some interest. If we call n the number of errors 
equal to CD, Fig. 2, and N the number of errors between zero and m, 

the ratio ^ will be equal to the ratio of the areas ydx and O AB ; the 
latter being equal to J, we have 


the sum of the squares of the errors equal to x is nx"^^ or 


the sum of the squares of aU the errors comprised between zero and 
will be 


51^ I x^ydx', 

we will obtain the mean of these squares by their number 'E) then the 
mean quadratic error will be the square root of the quotient 5 calling 
this quantity u^ and substituting for y its value, we have 


2 / x'^^^^^dxx 

whence we find 

and w =2.4495 n. We have already m^'dy^ consequently, 



We know that in using tlie/(^) we find that the ratio of the error of 
the mean to the mean error is 



There is, as we see, a slight difference between the two values of this 
ratio. It results from this, that if the mean errors represent the same 
length in the/(^) and the formula we adopt, the square roots of the 
mean squares of the errors would not correspond exactly, and that to 
pass from one to the other it would be necessary, if we wish that the 
two formulas should really be functions of the same quantities, to adopt 

a mean value of — . It is evident that we could thus make equal the 

mean squares of the errors 5 but in this case the mean errors would not 
coincide; and that we would make use of a different right line than 
the one we have just employed, but, however, extremely near to it ; 
here would be but little advantage. We would find that the greatest 
differences would be equal to some thousandths. Although this diffi- 
culty is not serious, it may lead to some confusion. That is why, in 

*Tliis is obtained as follows. The value of the mean error, y, from the note to 
Article III, is y= = . If we call wi the value of u when 2/=^=e~^'*', we have 

V'^ Jo 


We can integrate this by parts 5 


•Joo -Job 


' 1 -• 




1/^ Jo Vtt Jo 

«■ -2A= and«.=^. 



C. A. S. 


the precise calculatious the simi^ler way woiikl be to make use only of 
the mean error, it is sufficient foi- all tlie i)r('ceflin.i>- observations. 

Strangers appear to use almost exchisively the i)robable error; that 
is to say, tliat of which we have one chance out of two of not passing. 
This vahie we easily obtain by nmhing P equal to i in one of the 
formulas for the probability of not exceeding an error, x ; we find, by 

^1=0.8453: * 


with the right line we have 

from which we find 

m 2m^ *^ 

^=3(i_^ir) =0.8787. 


The difference 0.03 between the values of the ratio ^ occasions the 

same remarks as the preceding. The advantage which results fi'om the 
employment of the probable error is the more simple value of the proba- 
bility of not passing it ; it is easily remembered, but it has not a value 
more definite than that of any other error, only the numerical values of 
these probabilities are not so easy to remember. If, in working to 
hundredths, we take the mean error as that which we have 57 chances 
out of 100 of not j)assing, we have tor practice information of the same 
value as that which the knowledge of the probable error furnishes; the 
latter will have, with the formulas of this Memoir, the inconvenience of 
introducing some numerical coefficients less simple, and as the rapidity 
and ease of calculation are of much importance, we have thought it bet- 
ter to continue to make use of the mean error. 



The formula 

which represents the proT^ability of hitting a rectangle having its center 
at the mean point of impact, and its sides parallel to the axes of the 
target, being the product of the two probabilities j)^ and p., which we have 

See note to Article III.— C. A. S. 


of iu)t i>assing x aud 2/, we will have, calliDg z^ and £3 the errors of |)i and 
jjg, for the real value of the probability : 

the product £1 £3 can be neglected, since, after what we have vsaid, it can 
only represent some ten thousandths ; since P'= |>i^2 there remains for 
the appreciable part of the error attaching to P', 

A comparison of the values of P and P', in the preceding article shows 
that we obtain the greatest value of E when ^1 and 2)2 are each equal to 
0.99 ; the ratio of the half sides of the rectangle to the mean errors will 
then be 2.7 and E will be very nearly equal to —0.04. When the figure 
to be hit is very near this rectangle, and when we consider that this 
error is too large, we can make the indicated correction afterward, but 
only in cases where the method emplo^^ed permits us to consider that 
we have arrived at a result sufficiently exact to warrant this correction. 
In the case of the rectangle we will not have this inconvenience, for 
tables have been prepared for the values of the/ if). General Didion in 
his CaleuUis of Prohahilities applied to fire^ 1858, gives one very complete, 
but the argument is the mean square of the errors, a quantity which we 
rarely have given. [Others have been calculated, however, with more 
convenient arguments, and one is given at theend of M. Breger^s Memoir, 
but it has not been reproduced in this translation.] 



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